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Scattering Amplitudes and BCFW in $\mathcal{N}=2^*$ Theory

by Md. Abhishek, Subramanya Hegde, Dileep P. Jatkar, and Arnab Priya Saha

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Submission summary

Authors (as registered SciPost users): Md Abhishek · Subramanya Hegde · Arnab Saha
Submission information
Preprint Link: scipost_202203_00042v1  (pdf)
Date submitted: 2022-03-30 08:55
Submitted by: Abhishek, Md
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
Approach: Theoretical

Abstract

We use massive spinor helicity formalism to study scattering amplitudes in $\mathcal{N}=2^*$ super-Yang-Mills theory in four dimensions. We compute the amplitudes at an arbitrary point in the Coulomb branch of this theory. We compute amplitudes using projection from $\mathcal{N}=4$ theory and write three point amplitudes in a convenient form using special kinematics. We then compute four point amplitudes by carrying out massive BCFW shifts of the amplitudes. We find some of the shifted amplitudes have a pole at $z=\infty$. Taking the residue at $z=\infty$ into account ensures little group covariance of the final result.

Current status:
Has been resubmitted

Reports on this Submission

Report #2 by Anonymous (Referee 2) on 2022-4-29 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202203_00042v1, delivered 2022-04-29, doi: 10.21468/SciPost.Report.5004

Report

The paper studies scattering amplitudes in the Coulomb branch of four-dimensional N=2* supersymmetric theories. While the expressions for the amplitudes may be obtained by projection from N=4

The analysis in the paper highlights important structural differences between BCFW recursion in N=4 and N=2* theory. In particular, the appearance of little-group noncovariant terms in
intermediate steps of the computation. The cancellation of these terms in obtaining the final answer is an important consistency check of the method.

The paper is largely well written and the main results are presented clearly. Expectedly, these results would pave the way for a more exhaustive exploration of scattering amplitudes
in the moduli space of N=2* theories and more generally, N=2 theories.

We recommend the manuscript for publication, pending the addressal of a few comments, given below.

Requested changes

1. The authors may clarify if they are working with color ordered amplitudes.

2. The introduction could also be slightly expanded to provide general motivations for the study of N=2* theories, including further references to the literature if needed.

3. Some terms could also be clearly defined when used for the first time, e.g.
a. 'u-spinor' on page 3
b. 'band structure' in Section 3.4

4. On page 23, " In N = 4 SYM Coulomb branch, it was found that.." should have the relevant reference.

5. Some sentences could also be editorialized. For example:
a. In the Introduction: The sentence beginning with "We put long in quotes because.."
b. In the Conclusions: "Also, because the pole at infinity recursive structure of the amplitudes is nontrivial,"

  • validity: high
  • significance: good
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: good

Report #1 by Anonymous (Referee 1) on 2022-4-29 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202203_00042v1, delivered 2022-04-29, doi: 10.21468/SciPost.Report.4997

Strengths

The manuscript is technically very well written with proper review and explanations of all the ingredients required for the original amplitude computations

It contains mostly the essential references which makes it simpler to follow

Weaknesses

Lacks proper motivations for studying on-shell amplitudes in $\mathcal{N} = 2^*$ Super Yang-Mills

Report

In this paper the authors have studied tree level scattering amplitudes (in particular they have explicitly computed three and four point amplitudes) for $\mathcal{N} = 2^*$ super Yang-Mills (SYM) theory in the Coloumb branch, building up on works [arXiv : 1902.07204 & arXiv : 1902.07205] by Herderschee, Koren and Trott. The authors have used massive spinor-helicity variables and Britto-Cachazo-Feng-Witten (BCFW) formalism to compute the amplitudes. More preciously they use following two different methods.

1. Method of projection from $\mathcal{N} = 4$ SYM : Using this method three and four-point amplitudes have been computed in section 3 and section 4 of the paper. These amplitudes are the main results of this work. The three point amplitudes are also written in terms of $u$-spinors such that they can be conveniently used to build up BCFW recursion in section 5.

2. Massive super-BCFW recursion relation : In section 5, four point amplitudes are re-derived using $u$-spinor representations of the three point amplitudes mentioned above. This BCFW analysis is structurally different from $\mathcal{N} = 4$ SYM (although the method mentioned above shows the amplitudes can be obtained as projection from $\mathcal{N} = 4$ SYM), due to presence of 'pole at $\infty$'. In more detail, the massive BCFW shifts are $not$ covariant under the little group e.g. the integrand contains explicit BCFW shift parameter ($z$, in their notation). But for the particular example the authors are studying, they find that this 'non-covariance' is precisely canceled by the pole at $z = \infty$. This is not unexpected because the only way the recursion could work is when the two troublesome contributions to the amplitudes cancel each other. But this feature gives some technical advantages in the computation, since one can ignore both the contributions ($z$-dependent piece in the integrand and the residue at $z = \infty$) from the beginning.

The manuscript is technically very well written and it extends the applications of on-shell methods in computing tree-level amplitudes to larger class of supersymmetric theories. I recommend the manuscript for publication in SciPost Physics.

Requested changes

Here are some suggestions that, I believe, should improve the manuscript and make it more accessible to the readers - particularly working broadly in scattering amplitudes related topics.

I feel the paper lacks proper motivations. After briefly reviewing the success of on-shell amplitude program in $\mathcal{N} = 4$ SYM, it somewhat abruptly starts discussing about $\mathcal{N} = 2^*$ theory (in the third paragraph of the introduction) as : "Another place where this generalised spinor helicity formalism can be used is in studying amplitudes in the $\mathcal{N} = 2^*$ theory ...". The authors may consider adding a paragraph there explaining why one should be interested in studying $\mathcal{N} = 2^*$ SYM amplitudes in the first place (e.g. describing if there are some interesting issues/results that can be tackled/checked by studying these amplitudes). In the beginning of section 5, possible connections with massive amplitudes in $\mathcal{N} = 2$ SYM have been mentioned. This part may also be expanded in the introduction.

  • validity: good
  • significance: ok
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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